A152149 Decimal expansion of the angle B in the doubly golden triangle ABC.

6, 5, 7, 4, 0, 5, 4, 8, 2, 9, 7, 6, 5, 3, 2, 5, 9, 2, 3, 8, 0, 9, 6, 8, 5, 4, 1, 5, 2, 9, 3, 9, 7, 1, 2, 6, 5, 4, 1, 4, 9, 5, 9, 4, 6, 4, 8, 7, 8, 3, 9, 3, 7, 0, 7, 8, 2, 0, 9, 2, 8, 0, 8, 5, 8, 8, 5, 3, 9, 5, 0, 6, 1, 3, 8, 1, 7, 7, 3, 5, 0, 7, 0, 1, 7, 1, 5, 1, 6, 5, 4, 4, 0, 5, 2, 2, 7, 8, 0, 5, 2, 8, 1, 2, 6

Offset

0,1

Comments

There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t. (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)

For doubly silver and doubly e-ratio triangles, see A188543 and A188544.

For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

References

Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.

Links

Iain Fox, Table of n, a(n) for n = 0..20000

Jordi Dou, Clark Kimberling and Laurence Kuipers, A Fibonacci sequence of nested triangles, Problem S29, Amer. Math. Monthly 89 (1982) 696-697; proposed 87 (1980) 302.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Formula

B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.

Example

The number B begins with 0.65740548 (equivalent to 37.666559... degrees).

Mathematica

r = (1 + 5^(1/2))/2; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

Prog

(PARI) t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ Iain Fox, Feb 11 2020

Crossrefs

Cf. A000045, A188543, A188544, A376961 (length of shortest side of the doubly golden triangle that has area 1).

Sequence in context: A201330 A089826 A211172 * A338287 A365927 A196400

Adjacent sequences: A152146 A152147 A152148 * A152150 A152151 A152152

Keyword

nonn,cons,changed

Author

Clark Kimberling, Nov 26 2008

Extensions

Keyword:cons added and offset corrected by R. J. Mathar, Jun 18 2009

Status

approved